We consider random walk and self-avoiding walk whose 1-step distribution is given by D, and oriented percolation whose bond-occupation probability is proportional to D. Suppose that D(x) decays as |x|−d − α with α > 0. For random walk in any dimension d and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension dc ≡ 2(α ∧ 2), we prove large-t asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length t or the average spatial size of an oriented percolation cluster at time t. This proves the conjecture for long-range self-avoiding walk in [Ann. Inst. H. Poincaré Probab. Statist. (2010), to appear] and for long-range oriented percolation in [Probab. Theory Related Fields 142 (2008) 151–188] and [Probab. Theory Related Fields 145 (2009) 435–458].
"Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation." Ann. Probab. 39 (2) 507 - 548, March 2011. https://doi.org/10.1214/10-AOP557